The confetti bag probability

Probability Level 3

20 people each get a bag with 1-10 pieces of confetti of a RANDOMLY SELECTED color (number of pieces of confetti selected RANDOMLY). The confetti is either ALL blue or ALL red in any one bag. When they have all received their bags they are all organized into 10 groups of 2 randomly and share their color and number of pieces of confetti with each other.

What is the probability of EXACTLY ONE of the pairs of people having bags with the same color and number of pieces of confetti?

It's impossible to calculate About 1 in 5 About 1 in 10 About 25% 0.615 About 1 in 3 About 80% About 1 in 50

1 solution

Brian Charlesworth
Jul 15, 2018

For any given pair, the probability that they have the same number of pieces of confetti in their bags is $\dfrac{1}{10}$ , and that the pieces are the same colour is $\dfrac{1}{2}$ , and thus the probability that the number of pieces and colour are the same is $\dfrac{1}{10} \times \dfrac{1}{2} = \dfrac{1}{20}$ .

Then for any given pair, the probability that either the colour and/or number of their pieces differ is $1 - \dfrac{1}{20} = \dfrac{19}{20}$ .

For any way of pairing off the 20 people, there are 10 ways where precisely one of the pairs has matching bags of confetti, with this pair occurring with probability $\dfrac{1}{20}$ and the other 9 pairs each occurring with probability $\dfrac{19}{20}$ . The desired probability is then

$10 \times \left(\dfrac{1}{20}\right) \times \left(\dfrac{19}{20}\right)^{9} = \dfrac{1}{2}\left(\dfrac{19}{20}\right)^{9} \approx 0.315$ , or about $\boxed{\text{about 1 in 3}}$ .

The confetti bag probability

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